The origin of water anomalies hides in an experimentally inaccessible region of the phase diagram known as no-man's land, bounded at low temperature by the domain of stability of amorphous glasses, and at high temperature by the homogeneous nucleation line, below which liquid water loses its metastability. The existence of at least two different forms of glass on one side, i.e., the low-density amorphous (LDA) and the high-density amorphous (HDA) ices, and of one anomalous liquid on the other side, points to a hidden connection between these states, whose understanding has the potential to uncover what happens in no-man's land and shed light on the complex nature of water's behavior. Here, we develop a Neural Network scheme capable of discerning local structures beyond tetrahedrality. Applied over a wide region of the water's phase diagram, we show that the local structures that characterize both LDA and HDA amorphous phases are indeed embedded in the supercooled liquid phase. Remarkably, the rapid increase in the LDA-like population with supercooling occurs in the same temperature and pressure region where thermodynamic fluctuations are maximized, linking these structures with water's anomalies. At the same time, the population of HDA-like environments rapidly increases with pressure, becoming the majority component at high density. Our results show that both LDA and HDA are genuine glasses, and provide a microscopic connection between the non-equilibrium and equilibrium phase diagrams of water.